Question

The mean of the following series is . (rounded off to three decimal places)
\(0.01,\ 0.02,\ 0.03,\ \ldots,\ 1\)

Hint:

This is an AP of 100 terms running from 0.01 to 1; the mean of an AP equals the average of its first and last terms.

Answer 1

Correct Answer: 0.505

Instead of using the symmetry shortcut, let's find the mean the direct way: add up every term and divide by how many terms there are.

The series $0.01, 0.02, 0.03, \ldots, 1$ is an arithmetic progression. Multiply every term by 100 to work with whole numbers: it becomes $1, 2, 3, \ldots, 100$, which is just the first 100 natural numbers scaled down by a factor of 100.

The sum of the first $n$ natural numbers is a standard result: $S_n = \frac{n(n+1)}{2}$. Here $n = 100$, so

\[ S_{100} = \frac{100 \times 101}{2} = 5050 \]

This is the sum of the scaled series $1, 2, \ldots, 100$. Scale back down by dividing by 100 to get the sum of the original series:

\[ \text{Sum of original series} = \frac{5050}{100} = 50.5 \]

Now divide by the number of terms, $n = 100$, to get the mean:

\[ \text{Mean} = \frac{50.5}{100} = 0.505 \]

Let's summarize:

• Scaling the series by 100 turns it into the first 100 natural numbers, whose sum is given by a known formula.
• Scaling the sum back down and dividing by 100 terms gives the mean directly, without needing the average of endpoints shortcut.

So the mean, rounded to three decimal places, is $0.505$.